4.10 - Antiderivatives. Antiderivatives Definition A function F is called an antiderivative of f if F ′(x) = f (x) for all x on an interval I. Theorem.

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Presentation transcript:

Antiderivatives

Antiderivatives Definition A function F is called an antiderivative of f if F ′(x) = f (x) for all x on an interval I. Theorem If F is an antiderivative of f on an interval I, then the most general antiderivative of f on I is the family of functions given by F(x) + c, where c is an arbitrary constant. F(x) + c is a called a family of functions (or antiderivatives).

Examples Determine the general antiderivative of the following functions.

Examples Determine f if …

Example A particle is moving according to the function a(t) = cos t + sin t where s(0) = 0 and v(0) = 5. Find the position of the particle.

Example The graph of a derivative of some function is given below. Sketch a possible graph of the function.

Slope (or Directional) Fields A slope (or directional) field is a way of graphically representing a family of antiderivatives. Slope Field Generator Example: Use the directional field generator to graph the antiderivative that satisfies F(0) = 0 if